Linear and projective boundary of nilpotent groups

TL;DR

This paper introduces a new boundary concept for metric spaces based on a pseudometric on unbounded subsets, and characterizes these boundaries for nilpotent groups, showing they are quasi-isometric invariants.

Contribution

It defines a novel boundary framework using a pseudometric guided by sublinear closeness, and characterizes boundaries of nilpotent groups as disjoint unions of spheres or projective spaces.

Findings

Boundaries are quasi-isometric invariants.

Boundaries of nilpotent groups are disjoint unions of spheres or projective spaces.

Applicable to polynomial growth graphs and random walks.

Abstract

We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub-(semi)-groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.